Existence of positive solution for a class of nonlocal elliptic problems in the half space with a hole

TL;DR

This paper proves the existence of positive solutions for a class of nonlocal fractional elliptic problems with a hole in a half-space, using variational and topological methods.

Contribution

It establishes existence results for nonlocal elliptic equations with fractional Laplacian in a half-space with a hole, extending previous work to this geometric setting.

Findings

Existence of positive solutions under specified conditions.

Application of variational and topological methods.

Results for fractional Laplacian problems in complex domains.

Abstract

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{eqnarray}\label{eq:0.1} &&\left\{\begin{array}{l} (-\Delta)^{s} u+u=|u|^{p-2} u \text { in } \Omega_{r} \\ u \geq 0 \quad \text { in }\Omega_{r} \text { and } u \neq 0 \\ u=0 \quad \mathbb{R}^{N} \backslash \Omega_{r} \end{array}\right., \end{eqnarray} involving the fractional Laplacian operator $(-\Delta)^{s},$ where $s \in(0,1), N>2 s$, $\Omega_{r}$ is the half space with a hole in $\mathbb{R}^N$ and $p \in\left(2,2_{s}^{*}\right) .$ The main technical approach is based on variational and topological methods.